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Full text of "Mathematical And Physical Papers - Iii"

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L, is regularly refracted at S, regularly reflected at T, and scattered on emergence at P. The lines LS, ST, TP will evidently lie in the plane LLQP. Let $, ' be the angles of incidence and refraction at 8. We have
= csec<j[>-f 2p.t sec
c tan
2$ tan   ' =
sn    =
' ...... (18).
The obliquity being supposed small, we may expand, and retain only the squares of small quantities, the terms thus neglected involving only fourth and higher powers. We get in the first place from (17)
R^c + tyt + h + bfa + ZpA^ + j^ ......... (19).
But equations (18) give
and substituting in (19) we get
To obtain jR2, it will be sufficient to interchange c and h, s and u, since if we supposed the course of the ray reversed it would emanate from E} be regularly refracted and reflected, then scattered on emergence at P, and so would reach L. Interchanging, subtracting, and reducing, we obtain
This formula is more general than (16), since no approximation has yet been made depending on the magnitude of t. In practice, however, t is actually small compared with c and k, so that we may simplify the formula by retaining only the first power of t, which reduces (20) to (16), inasmuch as
and s2 = (x - a)2 + (y - Z>)2.
11. Let us now proceed to apply the formula (16) to the explanation of the phenomena. In discussing this formula, it is to be remembered that x, y are the co-ordinates of the point of the mirror on which a fringe is seen projected. Since the direction